Branching random walk with selection at critical rate

Bastien Mallein

Published 2015-02-25Version 1

We consider a branching-selection particle system on the real line. In this model the total size of the population at time $n$ is limited by $e^{a n^{1/3}}$. At each step $n$, every individual dies while reproducing independently, making children around their current position according to i.i.d. point processes. Only the $e^{a(n+1)^{1/3}}$ rightmost children survive to form the $(n+1)^\mathrm{th}$ generation. This process can be seen as a generalisation of the branching random walk with selection of the $N$ rightmost individuals at each time, introduced by Brunet and Derrida. We obtain the asymptotic behaviour of position of the extremal particles alive at time $n$ by coupling this process with a branching random walk with a killing boundary.